manipulated, has been invented by Dr Whitehead for the purpose of
showing how points might be manufactured from sense-data. We have first
of all to observe that there are no infinitesimal sense-data: any
surface we can see, for example, must be of some finite extent. But what
at first appears as one undivided whole is often found, under the
influence of attention, to split up into parts contained within the
whole. Thus one spatial object may be contained within another, and
entirely enclosed by the other. This relation of enclosure, by the help
of some very natural hypotheses, will enable us to define a "point" as a
certain class of spatial objects, namely all those (as it will turn out
in the end) which would naturally be said to contain the point. In order
to obtain a definition of a "point" in this way, we proceed as follows:
Given any set of volumes or surfaces, they will not in general converge
into one point. But if they get smaller and smaller, while of any two of
the set there is always one that encloses the other, then we begin to
have the kind of conditions which would enable us to treat them as
having a point for their limit. The hypotheses required for the relation
of enclosure are that (1) it must be transitive; (2) of two _different_
spatial objects, it is impossible for each to enclose the other, but a
single spatial object always encloses itself; (3) any set of spatial
objects such that there is at least one spatial object enclosed by them
all has a lower limit or minimum, _i.e._ an object enclosed by all of
them and enclosing all objects which are enclosed by all of them; (4) to
prevent trivial exceptions, we must add that there are to be instances
of enclosure, _i.e._ there are really to be objects of which one
encloses the other. When an enclosure-relation has these properties, we
will call it a "point-producer." Given any relation of enclosure, we
will call a set of objects an "enclosure-series" if, of any two of them,
one is contained in the other. We require a condition which shall secure
that an enclosure-series converges to a point, and this is obtained as
follows: Let our enclosure-series be such that, given any other
enclosure-series of which there are members enclosed in any arbitrarily
chosen member of our first series, then there are members of our first
series enclosed in any arbitrarily chosen member of our second series.
In this case, our first enclosure-series may be called a "punctual
enclosure-series." Then a "point" is all the objects which enclose
members of a given punctual enclosure-series. In order to ensure
infinite divisibility, we require one further property to be added to
those defining point-producers, namely that any object which encloses
itself also encloses an object other than itself. The "points" generated
by point-producers with this property will be found to be such as
geometry requires.
(3) The question of time, so long as we confine ourselves to one private
world, is rather less complicated than that of space, and we can see
pretty clearly how it might be dealt with by such methods as we have
been considering. Events of which we are conscious do not last merely
for a mathematical instant, but always for some finite time, however
short. Even if there be a physical world such as the mathematical theory
of motion supposes, impressions on our sense-organs produce sensations
which are not merely and strictly instantaneous, and therefore the
objects of sense of which we are immediately conscious are not strictly
instantaneous. Instants, therefore, are not among the data of
experience, and, if legitimate, must be either inferred or constructed.
It is difficult to see how they can be validly inferred; thus we are
left with the alternative that they must be constructed. How is this to
be done?
Immediate experience provides us with two time-relations among events:
they may be simultaneous, or one may be earlier and the other later.
These two are both part of the crude data; it is not the case that only
the events are given, and their time-order is added by our subjective
activity. The time-order, within certain limits, is as much given as the
events. In any story of adventure you will find such passages as the
following: "With a cynical smile he pointed the revolver at the breast
of the dauntless youth. 'At the word _three_ I shall fire,' he said. The
words one and two had already been spoken with a cool and deliberate
distinctness. The word _three_ was forming on his lips. At this moment a
blinding flash of lightning rent the air." Here we have
simultaneity--not due, as Kant would have us believe, to the subjective
mental apparatus of the dauntless youth, but given as objectively as the
revolver and the lightning. And it is equally given in immediate
experience that the words _one_ and _two_ come earlier than the flash.
These time-relations hold between events which are not strictly
instantaneous. Thus one event may begin sooner than another, and
therefore be before it, but may continue after the other has begun, and
therefore be also simultaneous with it. If it persists after the other
is over, it will also be later than the other. Earlier, simultaneous,
and later, are not inconsistent with each other when we are concerned
with events which last for a finite time, however short; they only
become inconsistent when we are dealing with something instantaneous.
It is to be observed that we cannot give what may be called _absolute_
dates, but only dates determined by events. We cannot point to a time
itself, but only to some event occurring at that time. There is
therefore no reason in experience to suppose that there are times as
opposed to events: the events, ordered by the relations of simultaneity
and succession, are all that experience provides. Hence, unless we are
to introduce superfluous metaphysical entities, we must, in defining
what mathematical physics can regard as an instant, proceed by means of
some construction which assumes nothing beyond events and their temporal
relations.
If we wish to assign a date exactly by means of events, how shall we
proceed? If we take any one event, we cannot assign our date exactly,
because the event is not instantaneous, that is to say, it may be
simultaneous with two events which are not simultaneous with each other.
In order to assign a date exactly, we must be able, theoretically, to
determine whether any given event is before, at, or after this date, and
we must know that any other date is either before or after this date,
but not simultaneous with it. Suppose, now, instead of taking one event
A, we take two events A and B, and suppose A and B partly overlap, but B
ends before A ends. Then an event which is simultaneous with both A and
B must exist during the time when A and B overlap; thus we have come
rather nearer to a precise date than when we considered A and B alone.
Let C be an event which is simultaneous with both A and B, but which
ends before either A or B has ended. Then an event which is simultaneous
with A and B and C must exist during the time when all three overlap,
which is a still shorter time. Proceeding in this way, by taking more
and more events, a new event which is dated as simultaneous with all of
them becomes gradually more and more accurately dated. This suggests a
way by which a completely accurate date can be defined.
A____________________
B____________________
C________
Let us take a group of events of which any two overlap, so that there is
some time, however short, when they all exist. If there is any other
event which is simultaneous with all of these, let us add it to the
group; let us go on until we have constructed a group such that no event
outside the group is simultaneous with all of them, but all the events
inside the group are simultaneous with each other. Let us define this
whole group as an instant of time. It remains to show that it has the
properties we expect of an instant.
What are the properties we expect of instants? First, they must form a
series: of any two, one must be before the other, and the other must be
not before the one; if one is before another, and the other before a
third, the first must be before the third. Secondly, every event must be
at a certain number of instants; two events are simultaneous if they are
at the same instant, and one is before the other if there is an instant,
at which the one is, which is earlier than some instant at which the
other is. Thirdly, if we assume that there is always some change going
on somewhere during the time when any given event persists, the series
of instants ought to be compact, _i.e._ given any two instants, there
ought to be other instants between them. Do instants, as we have defined
them, have these properties?
We shall say that an event is "at" an instant when it is a member of the
group by which the instant is constituted; and we shall say that one
instant is before another if the group which is the one instant contains
an event which is earlier than, but not simultaneous with, some event in
the group which is the other instant. When one event is earlier than,
but not simultaneous with another, we shall say that it "wholly
precedes" the other. Now we know that of two events which are not
simultaneous, there must be one which wholly precedes the other, and in
that case the other cannot also wholly precede the one; we also know
that, if one event wholly precedes another, and the other wholly
precedes a third, then the first wholly precedes the third. From these
facts it is easy to deduce that the instants as we have defined them
form a series.
We have next to show that every event is "at" at least one instant,
_i.e._ that, given any event, there is at least one class, such as we
used in defining instants, of which it is a member. For this purpose,
consider all the events which are simultaneous with a given event, and
do not begin later, _i.e._ are not wholly after anything simultaneous
with it. We will call these the "initial contemporaries" of the given
event. It will be found that this class of events is the first instant
at which the given event exists, provided every event wholly after some
contemporary of the given event is wholly after some _initial_
contemporary of it.
Finally, the series of instants will be compact if, given any two events
of which one wholly precedes the other, there are events wholly after
the one and simultaneous with something wholly before the other. Whether
this is the case or not, is an empirical question; but if it is not,
there is no reason to expect the time-series to be compact.[17]
[17] The assumptions made concerning time-relations in the above are
as follows:--
I. In order to secure that instants form a series, we assume:
(a) No event wholly precedes itself. (An "event" is defined as
whatever is simultaneous with something or other.)
(b) If one event wholly precedes another, and the other wholly
precedes a third, then the first wholly precedes the third.
(c) If one event wholly precedes another, it is not simultaneous
with it.
(d) Of two events which are not simultaneous, one must wholly
precede the other.
II. In order to secure that the initial contemporaries of a given
event should form an instant, we assume:
(e) An event wholly after some contemporary of a given event is
wholly after some _initial_ contemporary of the given event.
III. In order to secure that the series of instants shall be
compact, we assume:
(f) If one event wholly precedes another, there is an event wholly
after the one and simultaneous with something wholly before the
other.
This assumption entails the consequence that if one event covers the
whole of a stretch of time immediately preceding another event, then
it must have at least one instant in common with the other event;
_i.e._ it is impossible for one event to cease just before another
begins. I do not know whether this should be regarded as inadmissible.
For a mathematico-logical treatment of the above topics, _cf._ N.
Wilner, "A Contribution to the Theory of Relative Position," _Proc.
Camb. Phil. Soc._, xvii. 5, pp. 441-449.
Thus our definition of instants secures all that mathematics requires,
without having to assume the existence of any disputable metaphysical
entities.
Instants may also be defined by means of the enclosure-relation, exactly
as was done in the case of points. One object will be temporally
enclosed by another when it is simultaneous with the other, but not
before or after it. Whatever encloses temporally or is enclosed
temporally we shall call an "event." In order that the relation of
temporal enclosure may be a "point-producer," we require (1) that it
should be transitive, _i.e._ that if one event encloses another, and the
other a third, then the first encloses the third; (2) that every event
encloses itself, but if one event encloses another different event, then
the other does not enclose the one; (3) that given any set of events
such that there is at least one event enclosed by all of them, then
there is an event enclosing all that they all enclose, and itself
enclosed by all of them; (4) that there is at least one event. To ensure
infinite divisibility, we require also that every event should enclose
events other than itself. Assuming these characteristics, temporal
enclosure is an infinitely divisible point-producer. We can now form an
"enclosure-series" of events, by choosing a group of events such that of
any two there is one which encloses the other; this will be a "punctual
enclosure-series" if, given any other enclosure-series such that every
member of our first series encloses some member of our second, then
every member of our second series encloses some member of our first.
Then an "instant" is the class of all events which enclose members of a
given punctual enclosure-series.
The correlation of the times of different private worlds so as to
produce the one all-embracing time of physics is a more difficult
matter. We saw, in Lecture III., that different private worlds often
contain correlated appearances, such as common sense would regard as
appearances of the same "thing." When two appearances in different
worlds are so correlated as to belong to one momentary "state" of a
thing, it would be natural to regard them as simultaneous, and as thus
affording a simple means of correlating different private times. But
this can only be regarded as a first approximation. What we call one
sound will be heard sooner by people near the source of the sound than
by people further from it, and the same applies, though in a less
degree, to light. Thus two correlated appearances in different worlds
are not necessarily to be regarded as occurring at the same date in
physical time, though they will be parts of one momentary state of a
thing. The correlation of different private times is regulated by the
desire to secure the simplest possible statement of the laws of physics,
and thus raises rather complicated technical problems; but from the
point of view of philosophical theory, there is no very serious
difficulty of principle involved.
The above brief outline must not be regarded as more than tentative and
suggestive. It is intended merely to show the kind of way in which,
given a world with the kind of properties that psychologists find in the
world of sense, it may be possible, by means of purely logical
constructions, to make it amenable to mathematical treatment by defining
series or classes of sense-data which can be called respectively
particles, points, and instants. If such constructions are possible,
then mathematical physics is applicable to the real world, in spite of
the fact that its particles, points, and instants are not to be found
among actually existing entities.
The problem which the above considerations are intended to elucidate is
one whose importance and even existence has been concealed by the
unfortunate separation of different studies which prevails throughout
the civilised world. Physicists, ignorant and contemptuous of
philosophy, have been content to assume their particles, points, and
instants in practice, while conceding, with ironical politeness, that
their concepts laid no claim to metaphysical validity. Metaphysicians,
obsessed by the idealistic opinion that only mind is real, and the
Parmenidean belief that the real is unchanging, repeated one after
another the supposed contradictions in the notions of matter, space, and
time, and therefore naturally made no endeavour to invent a tenable
theory of particles, points, and instants. Psychologists, who have done
invaluable work in bringing to light the chaotic nature of the crude
materials supplied by unmanipulated sensation, have been ignorant of
mathematics and modern logic, and have therefore been content to say
that matter, space, and time are "intellectual constructions," without
making any attempt to show in detail either how the intellect can
construct them, or what secures the practical validity which physics
shows them to possess. Philosophers, it is to be hoped, will come to
recognise that they cannot achieve any solid success in such problems
without some slight knowledge of logic, mathematics, and physics;
meanwhile, for want of students with the necessary equipment, this vital
problem remains unattempted and unknown.
There are, it is true, two authors, both physicists, who have done
something, though not much, to bring about a recognition of the problem
as one demanding study. These two authors are Poincare and Mach,
Poincare especially in his _Science and Hypothesis_, Mach especially in
his _Analysis of Sensations_. Both of them, however, admirable as their
work is, seem to me to suffer from a general philosophical bias.
Poincare is Kantian, while Mach is ultra-empiricist; with Poincare
almost all the mathematical part of physics is merely conventional,
while with Mach the sensation as a mental event is identified with its
object as a part of the physical world. Nevertheless, both these
authors, and especially Mach, deserve mention as having made serious
contributions to the consideration of our problem.
When a point or an instant is defined as a class of sensible qualities,
the first impression produced is likely to be one of wild and wilful
paradox. Certain considerations apply here, however, which will again be
relevant when we come to the definition of numbers. There is a whole
type of problems which can be solved by such definitions, and almost
always there will be at first an effect of paradox. Given a set of
objects any two of which have a relation of the sort called "symmetrical
and transitive," it is almost certain that we shall come to regard them
as all having some common quality, or as all having the same relation to
some one object outside the set. This kind of case is important, and I
shall therefore try to make it clear even at the cost of some repetition
of previous definitions.
A relation is said to be "symmetrical" when, if one term has this
relation to another, then the other also has it to the one. Thus
"brother or sister" is a "symmetrical" relation: if one person is a
brother or a sister of another, then the other is a brother or sister of
the one. Simultaneity, again, is a symmetrical relation; so is equality
in size. A relation is said to be "transitive" when, if one term has
this relation to another, and the other to a third, then the one has it
to the third. The symmetrical relations mentioned just now are also
transitive--provided, in the case of "brother or sister," we allow a
person to be counted as his or her own brother or sister, and provided,
in the case of simultaneity, we mean complete simultaneity, _i.e._
beginning and ending together.
But many relations are transitive without being symmetrical--for
instance, such relations as "greater," "earlier," "to the right of,"
"ancestor of," in fact all such relations as give rise to series. Other
relations are symmetrical without being transitive--for example,
difference in any respect. If A is of a different age from B, and B of a
different age from C, it does not follow that A is of a different age
from C. Simultaneity, again, in the case of events which last for a
finite time, will not necessarily be transitive if it only means that
the times of the two events overlap. If A ends just after B has begun,
and B ends just after C has begun, A and B will be simultaneous in this
sense, and so will B and C, but A and C may well not be simultaneous.
All the relations which can naturally be represented as equality in any
respect, or as possession of a common property, are transitive and
symmetrical--this applies, for example, to such relations as being of
the same height or weight or colour. Owing to the fact that possession
of a common property gives rise to a transitive symmetrical relation, we
come to imagine that wherever such a relation occurs it must be due to a
common property. "Being equally numerous" is a transitive symmetrical
relation of two collections; hence we imagine that both have a common
property, called their number. "Existing at a given instant" (in the
sense in which we defined an instant) is a transitive symmetrical
relation; hence we come to think that there really is an instant which
confers a common property on all the things existing at that instant.
"Being states of a given thing" is a transitive symmetrical relation;
hence we come to imagine that there really is a thing, other than the
series of states, which accounts for the transitive symmetrical
relation. In all such cases, the class of terms that have the given
transitive symmetrical relation to a given term will fulfil all the
formal requisites of a common property of all the members of the class.
Since there certainly is the class, while any other common property may
be illusory, it is prudent, in order to avoid needless assumptions, to
substitute the class for the common property which would be ordinarily
assumed. This is the reason for the definitions we have adopted, and
this is the source of the apparent paradoxes. No harm is done if there
are such common properties as language assumes, since we do not deny
them, but merely abstain from asserting them. But if there are not such
common properties in any given case, then our method has secured us
against error. In the absence of special knowledge, therefore, the
method we have adopted is the only one which is safe, and which avoids
the risk of introducing fictitious metaphysical entities.
LECTURE V
THE THEORY OF CONTINUITY
The theory of continuity, with which we shall be occupied in the present
lecture, is, in most of its refinements and developments, a purely
mathematical subject--very beautiful, very important, and very
delightful, but not, strictly speaking, a part of philosophy. The
logical basis of the theory alone belongs to philosophy, and alone will
occupy us to-night. The way the problem of continuity enters into
philosophy is, broadly speaking, the following: Space and time are
treated by mathematicians as consisting of points and instants, but they
also have a property, easier to feel than to define, which is called
continuity, and is thought by many philosophers to be destroyed when
they are resolved into points and instants. Zeno, as we shall see,
proved that analysis into points and instants was impossible if we
adhered to the view that the number of points or instants in a finite
space or time must be finite. Later philosophers, believing infinite
number to be self-contradictory, have found here an antinomy: Spaces and
times could not consist of a _finite_ number of points and instants, for
such reasons as Zeno's; they could not consist of an _infinite_ number
of points and instants, because infinite numbers were supposed to be
self-contradictory. Therefore spaces and times, if real at all, must not
be regarded as composed of points and instants.
But even when points and instants, as independent entities, are
discarded, as they were by the theory advocated in our last lecture, the
problems of continuity, as I shall try to show presently, remain, in a
practically unchanged form. Let us therefore, to begin with, admit
points and instants, and consider the problems in connection with this
simpler or at least more familiar hypothesis.